What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined?
Answer: Instead of expanding the entire product, we can look only at terms that will multiply to give $x^2$.  We know that: $$x^2=x^2\cdot 1=x\cdot x=1\cdot x^2$$Knowing this, the $x^2$ term in the expansion will be the sum of these three terms: $$(-5x^2)(1)+(-7x)(-6x)+(1)(-x^2)$$Simplifying gives: \begin{align*}
(-5x^2)(1)+(-7x)(-6x)+(1)(-x^2)&=-5x^2+42x^2-x^2\\
&=\boxed{36}x^2
\end{align*}Consequently, the desired coefficient is $\boxed{36}$.